Item

Abstract

We give a criterion for the complete reducibility of modules satisfying a
composability condition for a meromorphic open-string vertex algebra $V$ using
the first cohomology of the algebra. For a $V$-bimodule $M$, let $\hat{H}^{1}_{\infty}(V, M)$ be the first cohomology of $V$ with the
coefficients in $M$. Let $\hat{Z}^{1}_{\infty}(V, M)$ be the subspace of $\hat{H}^{1}_{\infty}(V, M)$ canonically isomorphic to the space of derivations obtained from the zero mode of the right vertex operators of weight $1$
elements such that the difference between the skew-symmetric opposite action of
the left action and the right action on these elements are Laurent polynomials
in the variable. If $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$
for every $\Z$-graded $V$-bimodule $M$, then every left $V$-module satisfying a
composability condition is completely reducible. In particular, since a lower-bounded $\Z$-graded vertex algebra $V$ is a special meromorphic open-string vertex algebra and left $V$-modules are in fact what has been called generalized $V$-modules with lower-bounded weights (or lower-bounded generalized $V$-modules), this result provides a cohomological criterion for the complete reducibility of lower-bounded generalized modules for such a vertex algebra. We conjecture that the converse of the main theorem above is also true. We also prove that when a grading-restricted vertex algebra $V$ contains a subalgebra satisfying some familiar conditions, the composability
condition for grading-restricted generalized $V$-modules always holds and we
need $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$ only for every
$\Z$-graded $V$-bimodule $M$ generated by a grading-restricted subspace in our
complete reducibility theorem.